Ooh, a bite!
Right, so the first thing I consider non-negotiable is that every single human being has a right to live its life and they have a right to defend that life. Firearms are the most effective tool for self defense, so therefore I think everyone should be able to use them for legitimate self-defense.
That said, there are simply lousy people out there who want to do nothing but hurt others for their own benefit, and giving them such an effective tool as a firearm is just asking for trouble. To that end, my core stance on the matter is, "Good people aren't going to commit a crime with any firearm because good people don't commit crime in the first place. How do we ensure only good people have access to these tools?"
Background checks are the first line of defense. If you have a history of criminal violence or your local law enforcement has flagged you with an 'unstable' or similar flag (which can happen) then you shouldn't be able to purchase a firearm. Likewise, if you are mentally unstable or even on temporary medication that can make you unstable, you shouldn't be able to purchase a firearm (and hopefully a friend/family member has taken any that you own away).
Second, I think there should be VOLUNTARY (NOT mandatory) gun buyback programs offered nationwide. The huge number of firearms in this country does contribute to their ease-of-access, and buying them back would be a good way to cut down on any number of gun-related incidents in the nation. Combined with a mandatory background check system that has no loopholes, you'd see a good reduction in gun crime overall.
I oppose a gun registry and limitations on types of firearms or firearm accessories. A registry does nothing but show you don't trust the citizens who have passed background checks and frankly represents government intrusion into private life that we don't need or want in this nation. People buying guns second-hand aren't going to report that to a registry anyways, so it won't accomplish its primary function anyhow. "Scary black guns/assault weapons" are often identical to other firearms laid into more traditional wooden frames. Banning things like pistol grips, forward grips, types of stocks/braces, etc are just bans on guns and gear that makes those guns safer, easier, and frankly more fun to shoot. There is no reason why law-abiding citizens should be denied access to this gear.